Lyapunov stability analysis of covid 19 SIR modeling

Abstract

COVID-19 is an infectious disease which is spreading as a global pandemy to the whole world. This paper exlplores nonlinear compartemental dynamic models which is used to model the spread of Covid-19. The Susceptible-Infected-Removed (SIR) model is one of the compartmental dynamic models which can be used to simulate what happens when someone in the community catches a disease, like Covid-19. The classical SIR model assumes that the properties of individuals can be divided into three distinct compartments: S(t) is the total of susceptible persons, I(t) is the total of infected persons and R(t) is the total of recovered person from infection and are now immune against the disease. The model also includes the dedicated effort of the government, the decision makers, and the stakeholders. Theoretically, interventions such as social distancing, mass testing, and isolation of positive cases should slow the rate of the infection spreads. The analysis of equilibrium indicates that the model has two equilibriums. One of them isthe disease free equilibrium and the other one is the endemic equilibrium. If the effort level less than minimum level, than the spread of the virus becomes endemic and if the effort level more than minimum level, than the spread of the virus can be controlled. The basic reproduction number R0, is used to be an indicator of the expected number of individuals directly infected by an infectious person in a population where all individuals are susceptible to infection. If R0 < 1, then the disease free equilibrium point is stable, meaning that the virus spread can be controlled. If R0 > 1, then the disease free equilibrium point is unstable, meaning that the virus spread will continue. By constructing suitable Lyapunov function for SIR covid-19 model, the stability of the disease free equilibrium state of the model is thereby established.